To show that quadrilateral ABCD is a parallelogram, we need to demonstrate that both pairs of opposite sides are parallel. In coordinate geometry, two lines are parallel if their slopes are equal.

We have already calculated the slopes of the sides:

• Slope of AB: $5$
• Slope of BC: $\frac{1}{5}$
• Slope of CD: $5$
• Slope of DA: $\frac{1}{5}$

Verification:

1. Compare the slopes of AB and CD (opposite sides):

   • Slope of AB = 5
   • Slope of CD = 5

   Since the slopes are equal, AB is parallel to CD.

2. Compare the slopes of BC and DA (opposite sides):

   • Slope of BC = $\frac{1}{5}$
   • Slope of DA = $\frac{1}{5}$

   Since the slopes are equal, BC is parallel to DA.

Conclusion:

Since both pairs of opposite sides (AB and CD, BC and DA) are parallel, quadrilateral ABCD is indeed a parallelogram.

Would you like more details or have any further questions about this explanation?

Here are five related questions you might consider:

1. What are the properties of a parallelogram in terms of angles and sides?
2. How can you verify that ABCD is a rectangle?
3. How do you calculate the area of parallelogram ABCD using the vertices?
4. Can you find the coordinates of the midpoint of diagonals AC and BD? What do they reveal about the parallelogram?
5. How would you prove that a quadrilateral is a rhombus using slopes and distance?

Tip: In a parallelogram, the diagonals bisect each other, so the midpoints of the diagonals are the same.